The following Lemma is due to Rørdam ([1]}, Lemma 6.3), although we have expanded its statement slightly. Recall that a unital C\(^*\) -algebra \(A\)  is strongly \(\mathrm {K}_1\) -surjective if the canonical map
\(\mathcal {U}(B + \mathbb {C}(1_A)) \rightarrow \mathrm {K}_1(A)\) 
